15 problems

Let $C_{2k}$ be the cycle of length $2k$, and let $\operatorname{ex}^*(n,C_{2k})$ denote the maximum number of edges in a properly edge-colored $n$-vertex graph containing no rainb…

Let $C_{2k}$ be the cycle of length $2k$, and let ${\operatorname{ex}^\star}(n,C_{2k})$ denote the maximum number of edges in an edge-coloured graph on $n$ vertices with no properl…

Grzesik–Janzer–Nagy conjecture for even cycles. For every pair of integers $r,k\geq2$,

Let $C_{2k}$ denote the cycle of length $2k$, and let $\mathrm{ex}^*(n,H)$ be the maximum number of edges in a properly edge-coloured $n$-vertex graph containing no rainbow copy of…

Let $k$ be a positive integer, and let ${\mathcal C}_{\leq 2k}$ denote the family of all cycles of length at most $2k$. The Erdős girth conjecture. There is a constant $c(k)>0$ suc…

Let $C_{2k}$ be the cycle of length $2k$, let $C^d_{2k}$ denote its $d$-fold Cartesian product, and let $E_0\subseteq E(C^d_{2k})$ be a distance-3 matching, meaning that the distan…

Let $Q_n$ be the $n$-dimensional hypercube, let $C_{2l}$ denote the cycle of length $2l$, and let ${\rm ex}(Q_n,C_{2l})$ be the maximum number of edges in a $C_{2l}$-free spanning…

For an integer $r\geq 2$ and a graph $H$, let $f_r(n,H)$ be the smallest number $C$ such that some proper edge-colouring of $K_n$ with $C$ colours contains no $r$ vertex-disjoint c…

Let $Q_n$ be the hypercube graph, let $C_{2l}$ be the cycle of length $2l$, and let $e(Q_n)$ denote the number of edges of $Q_n$. For a graph $G$, write ${\rm ex}(G,H)$ for the max…

For positive integers $k$, let $C_{2k}$ denote the cycle of length $2k$, and let $\operatorname{ex}(n,C_{2k})$ be the maximum number of edges in an $n$-vertex graph containing no c…

Let $\ell\geqslant2$, and write … for the maximum number of edges in an $n$-vertex graph containing none of the cycles $C_3,C_4,\ldots,C_{2\ell}$. Erd s–Simonovits conjecture. … Th…

For an even cycle $C_{2\ell}$, let $\operatorname{ex}(n,C_{2\ell})$ denote its extremal number. Even-cycle extension conjecture. For every even cycle of length at least six, there…

Let $C_{2k}$ denote the cycle of length $2k$, and let $\mathrm{ex}(n,C_{2k})$ be the maximum number of edges in an $n$-vertex graph containing no copy of $C_{2k}$. Erdős's conjectu…

Let $\mathcal{G}$ be a graph with algebraic toric set $X$, and suppose it contains an even cycle $\mathcal{H}\cong\mathcal{C}_{2k}$. For each even cycle, choose variables correspon…

Let $X$ be the algebraic toric set associated to an even cycle $\mathcal{G}=\mathcal{C}_{2k}$, and let $ambda$ be the partition … For a partition $sigma=A\sqcup B$ of…